Numerical Maximization of the p-Laplacian Energy of a Two-Phase Material

Abstract

For a diffusion problem modeled by the p-Laplacian operator, we are interested in obtaining numerically the two-phase material which maximizes the internal energy. We assume that the amount of the best material is limited. In the framework of a relaxed formulation, we present two algorithms, a feasible directions method and an alternating minimization method. We show the convergence for both of them, and we provide an estimate for the error. Since for p > 2 both methods are only well-defined for a finite-dimensional approximation, we also study the difference between solving the finite-dimensional and the infinite-dimensional problems. Although the error bounds for both methods are similar, numerical experiments show that the alternating minimization method works better than the feasible directions one.